Question

I having difficulty integrating this
Ingrate (x^3)/sqrt(x^2 - 9)

I know that x= 3sec(theta)
Sqrt(x^2-9)= 3tan(theta)
x^3 = 27Sec^3(theta)

Once I put that all together I'm not sure what to do next.

Answer 1

so far, so good. But you forgot about dx:

dx = 3secθtanθ dθ

So, now you have

∫27sec^3(θ) / 3tanθ * 3secθtanθ
= 27∫sec^4(θ) dθ
= 27∫(tan^2(θ)+1) sec^2(θ) dθ
Now if you let u = tanθ, you have
= 27∫(u^2+1) du
= 27(u^3/3 + u)
= 27(tan^3(θ)/3 + tanθ)
= 9tanθ (tan^2(θ) + 3)

Now, the original substitutions,
tanθ = 1/3 √(x^2-9), so we wind up with

3√(x^2-9) (1/9 (x^2-9) + 3)
= 1/3 √(x^2-9) (x^2-9+27)
= 1/3 √(x^2-9) (x^2+18)